Average weights and power in weighted voting games (d4-journals-mss-2020-0007)

Answer: —

The anchor (named result in the paper)

Theorem 3 — class: algorithmic

PDF text-layer excerpt — equations and symbol order may be approximateIf W ∼Unif (∆n), then W ↓ k , k = 1, . . . , n, is distributed according to an absolutely continuous distri- bution supported on [1/n, 1] for k = 1 and on [0, 1/k] for k > 1, with piecewise polynomial density fn,k : [0, 1] →R given by: fn,k (x) := n(n −1) n −1 k −1 × min(n,⌊1/x⌋) X j=k (−1)j−k n −k j −k (1 −jx)n−2 .

statement extracted from the paper’s text layer; consult the paper for the exact wording

The continuous mirror question, as graded

Given \(\tau\), \(k\), rational \(x\), and a society with \(\tau\) top-choice types whose mass vector \(\mu\) is drawn uniformly from \(\Delta_\tau\), compute \(D_{\tau,k}(x)=\Pr_{\mu\sim\operatorname{Unif}(\Delta_\tau)}[\mu_k^\downarrow\ge x]\) exactly or within additive error \(\varepsilon\).

Packaged proof task (what the prover saw — nothing else)

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Prover output (gpt-6.1-sol / high, Bolzano prover prompt)

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Adversarial review (gpt-6.1-sol / medium, Bolzano verifier prompt)

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Tier: machine-proved, adversarially reviewed by an independent model. No human has checked this proof. Generated from pipeline artifacts by tools/atlas_build.py.